53edo: Difference between revisions
Tag: Reverted |
No need to remind readers of what a regular temperament is everywhere Tag: Undo |
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== Theory == | == Theory == | ||
53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]]. | 53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]]. | ||
As an equal temperament, it notably [[tempering out|tempers out]] [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma|schisma (32805/32768)]], [[15625/15552|kleisma (15625/15552)]], and [[amity comma|amity comma (1600000/1594323)]]. In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]] for which it is a [[Marvel#Tunings|relatively efficient tuning]], [[1728/1715|orwellisma (1728/1715)]], [[3125/3087|gariboh comma (3125/3087)]], and [[4375/4374|ragisma (4375/4374)]]. In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[marvel family #Athene|athene]] temperament. | |||
53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian|Arabic, Turkish, and Persian music]]. It can also be used as an extended [[3-limit|Pythagorean tuning]], since its fifths are almost indistinguishable from just. | 53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian|Arabic, Turkish, and Persian music]]. It can also be used as an extended [[3-limit|Pythagorean tuning]], since its fifths are almost indistinguishable from just. | ||
53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma. | 53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Linear temperaments === | === Linear temperaments === | ||
* [[List of edo-distinct 53et rank two temperaments]] | * [[List of edo-distinct 53et rank two temperaments]] | ||
* [[Schismic–Mercator equivalence continuum]] | * [[Schismic–Mercator equivalence continuum]] | ||